On function spaces and extension results for nonlocal Dirichlet problems
Bartªomiej Dyda
Wrocªaw University of Science and Technology, Poland
Let us consider the question, for which functions g : Rd \ Ω → R there is a function
u : Rd → R satisfying
Z
u(y) − u(x)
Lu(x) := p.v.
dy = 0
for x ∈ Ω ,
(1)
|y − x|1+2s
Rd
u(x) = g(x)
for x ∈ Rd \ Ω .
(2)
For open Ω, G ⊂ Rd dene two vector spaces by
2
Z Z
v(y)
−
v(x)
V (Ω, G) = v ∈ L2loc (Rd ) ∩ L2 (Ω)|
dx dy < ∞ ,
|x − y|d+2s
Ω G
c
HΩ (R ) = {v ∈ V (Ω, R )|v = 0 on Ω } .
d
d
Let us dene the notion of a variational solution.
Denition 1
(cf. Denition 2.5 in [1]). Let Ω ⊂ Rd be open and bounded. Let g ∈
V (Ω, Rd ). Then u ∈ V (Ω, Rd ) is called a variational solution to (1)(2), if u−g ∈ HΩ (Rd )
and for every ϕ ∈ HΩ (Rd )
Z Z
u(y) − u(x) ϕ(y) − ϕ(x)
dy dx = 0 .
(3)
|x − y|d+2s
Rd Rd
In [1] it is proved that such a variational solution u exists. However, in order to apply
Denition 1 one needs to prescribe the data function g in the vector space V (Ω, Rd ),
i.e. in particular one needs to prescribe all values of g in Rd . This leads to two obvious
questions:
Questions.
For which space of functions g : Ωc → R is there an extension operator
g 7→ ext(g) ∈ V (Ω, Rd )? Do elements of V (Ω, Rd ) have a trace in this space?
In the talk we will answer these questions.
Joint work with Moritz Kassmann (Universität Bielefeld).
References
[1] M. Felsinger, M. Kassmann, and P. Voigt. The Dirichlet problem for nonlocal operators.
279(3-4):779809, 2015.
,
Math. Z.